Accurate modeling of returns of financial assets and exogenous risk factors or endogenous risk factors, often referred to collectively as financial variables or just variables, for the purposes of risk estimation and portfolio optimization involves two essential distinct components: (1) modeling of the distribution of the stand-alone variables and (2) modeling the dependence structure between the variables.
In this discussion, the topic of modeling dependence between financial variables via a factor model is not considered but this does not limit the validity and generality of the discussion. Even in the case of a factor model, a multivariate assumption should be made for the joint distribution of the factors returns, which brings the discussion back to the current topic. Furthermore, this is not dependent on the nature of the factors—whether they are fundamentals, observable or latent statistical factors.
Point (1) above involves taking account of real world phenomena required to accurately model the stand-alone variables such as clustering of the volatility effect, heavy-tails of the distribution, and skeweness of the distribution. Many studies have been carried out in the last 40 years for the development of mathematical models for this purpose including the use of ARMA-GARCH processes and application of heavy-tailed and skewed distributions.
At the same time, the problems posed by point (2) have been largely neglected. Only in the last 15 years have significant efforts been dedicated to scientific research into the area of application of copula functions to model dependency between variables in the field of finance.
Only relatively recently have practitioners realized that modeling stand alone returns distributions and their dependencies are two distinct components of a complete model. Whereas in general, both modeling of the distributions of the stand alone variable and their dependencies are required to build an accurate multidimensional model for the returns of a set of financial variables.
Until fairly recently, predictive modeling of the future behavior of financial variables was achieved by assuming a given multidimensional distribution model, such as the multivariate normal or the multivariate Student's t distribution for example. This method leads to making two separate assumptions simultaneously. First, an assumption about the one-dimensional distribution of the variables viewed as stand-alone ones. And second, an assumption for the dependence between the variables. For instance, many practitioners will use the assumption of a multivariate normal as a model for the behavior of stock returns which in turn implies: (1) the normal distribution is the one dimensional model and (2) the covariance matrix is the description of the dependence structure between stock returns.
Among many practitioners, the covariance matrix is considered the standard and practical approach to dependency modeling. This practice dates back to at least the development of Modern Portfolio Theory by Harry Markowitz in the 1950's. The covariance matrix is still the standard in the induction for portfolio optimization and risk computation systems.
Any practical dependency model has to be flexible enough to account for several phenomena observed empirically in real-world financial data including that the dependence is nonlinear with changes in the values of financial variables. For example, that the dependence between asset prices becomes greater during periods of financial market stress is well known.
Additionally, in general, the dependency between financial variables is asymmetric. For example, most asset prices become relatively more dependent during significant financial market downturns than during good times.
The industry standard dependence model implied by the multivariate normal distribution fails to incorporate both of the aforementioned phenomena. The reasons for that include:                (i) First, the covariance matrix, which defines the dependence structure of financial returns, determines only linear dependencies, and the covariances are symmetric dependency measures.        (ii) Second, tail events are asymptotically independent under the multivariate normal (Gaussian) distribution        (iii) Third, the multivariate normal distributions describe only bivariate dependences.        
In modeling financial returns of a portfolio of assets that the large downturns (losses) are more dependent (associated) than large upturns (positive returns) is often observed, thus an asymmetry of the dependence structure between the asset returns is observed, which is in contradiction with reason (i).
If a large bivariate sample is generated using bivariate normal distribution with correlation close +1, meaning that two financial returns outcomes that are very positively correlated are modeled, the very large values (generated big losses, or big large returns) will be approximately independent, and that is the essence of reason (ii). That contradicts that in financial markets often very large losses in financial markets “go together”, that is a very big loss from one asset can lead to a very big loss of another one. This cannot be captured by modeling the asset returns with multivariate normal distribution, due to reason (ii).
Finally, very large losses go together in clusters, not just in the form of bivariate dependence, which contradicts reason (iii). One goal of the present invention is to described a copula-dependency model that is free of the deficiencies described above in reasons (i), (ii), and (iii).
From a historical perspective, modeling dependencies between financial variables advanced significantly in the 1990's. In theory, the most general possible way of describing dependence structure is through a copula function. For a vector of n variables, a copula function is an n-dimensional function satisfying certain properties, and which combines at least two one-dimensional distributions into a multivariate probability distribution.
Definition
Let d denote the number of securities, risk factors and residuals (the dimension), Id=[0, 1]d the d-dimensional unit hype-cube. A copula is a probability distribution on Id whose projections on each of the d axes are all uniform distributions on [0,1].
There is an existence and uniqueness result, which guarantees that any multivariate distribution has a unique copula behind the distribution under some regularity assumptions. Conversely, any copula function leads to a proper multivariate distribution, given a set of assumptions for the one-dimensional distributions. In effect, copula functions are the most general possible way of describing dependence. However, from a practical viewpoint these methods are often plagued by the difficulty in choosing a suitable copula function which is capable of describing the empirically observed phenomena in the data and at the same time be computationally feasible for use with large portfolios (for example, a large number of dimensions).